Abstract

In this work, the electric field emitted from a moving source, an electric point dipole, is analyzed for the purpose of illustrating the physics behind the Doppler effect. It is found that if the (translational) motion of the source is nonrelativistic, the Doppler effect is realized in two steps: the motion of the source first causes the dyadic Green function associated with the electric field to acquire an oscillation frequency in the far-field region of the source, and then the frequency leads to the Doppler effect. It is also demonstrated that the Doppler effect is observable only in the far-field region of the source.

© 2012 Optical Society of America

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References

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  1. The Doppler effect exists in different branches of physics. The present work is limited in the optical domain, and only focused on the change of frequency as a result of the motion of the source.
  2. B. Cairns and E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 8, 1922–1928 (1991).
    [CrossRef]
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
    [CrossRef]
  5. B. Rossi, Optics (Addison-Wesley, 1957).
  6. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  7. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, 1975).
  8. F. E. Low, Classical Field Theory—Electromagnetism and Gravitation (Wiley, 1997).
  9. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
    [CrossRef]
  10. O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
    [CrossRef]

2007 (1)

W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
[CrossRef]

1999 (1)

1998 (1)

O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
[CrossRef]

1991 (1)

Cairns, B.

Guo, W.

W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Keller, O.

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
[CrossRef]

O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, 1975).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, 1975).

Low, F. E.

F. E. Low, Classical Field Theory—Electromagnetism and Gravitation (Wiley, 1997).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Rossi, B.

B. Rossi, Optics (Addison-Wesley, 1957).

Wolf, E.

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
[CrossRef]

Phys. Rev. A (1)

O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
[CrossRef]

Other (6)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

B. Rossi, Optics (Addison-Wesley, 1957).

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, 1975).

F. E. Low, Classical Field Theory—Electromagnetism and Gravitation (Wiley, 1997).

The Doppler effect exists in different branches of physics. The present work is limited in the optical domain, and only focused on the change of frequency as a result of the motion of the source.

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Equations (16)

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j⃗(r⃗,t)=j⃗(r⃗1,t)δ(r⃗1r⃗)dr⃗1.
j⃗(r⃗,t)j⃗(r⃗1,t)δ(r⃗0r⃗)dr⃗1j⃗0(t)δ(r⃗0r⃗).
(21c22t2)E⃗(r⃗,t)·E⃗(r⃗,t)=μ0tj⃗(r⃗,t),
(2+ω2c2)E⃗(r⃗,ω)·E⃗(r⃗,ω)=iμ0ωj⃗(r⃗,ω).
χ⃗(r⃗,t)=12πχ⃗(r⃗,ω)eiωtdω,
χ⃗(r⃗,ω)=χ⃗(r⃗,t)eiωtdt.
G(r⃗,r⃗1)=(I+1k2)eik|r⃗r⃗1|4π|r⃗r⃗1|,
E⃗(r⃗,ω)=iμ0ωG(r⃗,r⃗1)·j⃗(r⃗1,ω)dr⃗1.
j⃗(r⃗,ω)=12πj⃗0(ω1)dω1δ(r⃗v⃗t)ei(ωω1)tdt,
E⃗(r⃗,ω)=iμ0ω2πei(ωω1)tdtG(r⃗,v⃗t)·j⃗0(ω1)dω1.
E⃗(r⃗,ω)=iμ0ω2πei(ωω1)tdtGR(r⃗,v⃗t)·j⃗0(ω1)dω1,
GR(r⃗,v⃗t)14π(I(r⃗v⃗t)(r⃗v⃗t)|r⃗v⃗t|2)eik|r⃗v⃗t||r⃗v⃗t|.
GR(r⃗,v⃗t)14π(Ir^r^)eik(|r⃗|v⃗·r^t)|r⃗|,
E⃗(r⃗,ω)=iμ0ω4π(1v⃗·r^c1)(Ir^r^)·j⃗0eik|r⃗||r⃗|δ(ωω01v⃗·r^c1),
GR(r⃗,v⃗t)14π(I(r⃗v⃗t)(r⃗v⃗t)|r⃗v⃗t|2)1|r⃗v⃗t|.
E⃗(r⃗,ω)=iμ0ω8π2ΩNei(ωω0)t(I(r⃗v⃗t)(r⃗v⃗t)|r⃗v⃗t|2)·j⃗01|r⃗v⃗t|dt,

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